Wave Properties

Types of waves

Type	Particle motion	Examples
Transverse	Perpendicular to wave direction	Light, string waves, surface water waves, seismic S-waves
Longitudinal	Parallel to wave direction	Sound, compression waves in spring, seismic P-waves
Mixed	Both	Water surface waves (orbital motion), surface seismic waves
Mechanical	Need a medium	Sound, water, springs
Electromagnetic	No medium needed	Light, radio, X-rays
Quantum (matter)	Probability waves	Electrons, atoms (de Broglie waves)

Four key properties

Property	Symbol	Description	Units
Wavelength	λ	Distance between successive peaks	m
Frequency	f	Cycles per second	Hz (= 1/s)
Period	T	Time for one cycle (= 1/f)	s
Amplitude	A	Maximum displacement from equilibrium	m, Pa, V/m, etc.
Speed	v	How fast wavefronts propagate	m/s
Wave number	k	2π/λ — radians per meter	rad/m
Angular frequency	ω	2π·f — radians per second	rad/s

The fundamental relation

v = λ · f

(speed = wavelength × frequency). Equivalent forms:

v = ω / k = λ · f = λ / T

Common wave parameters

Wave	Frequency	Wavelength	Speed
Earthquake P-wave (rock)	1-30 Hz	200-6000 m	~6000 m/s
Audible sound (air)	20-20,000 Hz	17 mm-17 m	~343 m/s
FM radio	88-108 MHz	~3 m	3 × 10⁸ m/s
Microwave (oven)	2.45 GHz	12 cm	3 × 10⁸ m/s
Visible red light	4.3 × 10¹⁴ Hz	700 nm	3 × 10⁸ m/s
Visible blue light	6.7 × 10¹⁴ Hz	450 nm	3 × 10⁸ m/s
X-ray	~10¹⁸ Hz	~1 nm	3 × 10⁸ m/s
Gamma ray	~10²⁰ Hz	~10⁻¹² m	3 × 10⁸ m/s

JavaScript — wave property calculations

// Wave equation v = λf
function waveSpeed(wavelength, frequency) { return wavelength * frequency; }
function frequency(speed, wavelength) { return speed / wavelength; }
function wavelength(speed, frequency) { return speed / frequency; }

// Speed of sound in air ≈ 343 m/s
console.log(`A4 = 440 Hz: λ = ${wavelength(343, 440).toFixed(2)} m`); // 0.78 m
console.log(`100 Hz bass: λ = ${wavelength(343, 100).toFixed(2)} m`); // 3.43 m
console.log(`20 kHz limit: λ = ${(wavelength(343, 20000) * 1000).toFixed(0)} mm`); // 17 mm

// Speed of light c
const c = 3e8;
console.log(`Red light: f = ${(frequency(c, 700e-9) / 1e12).toFixed(0)} THz`); // ~430
console.log(`Microwave 12cm: f = ${(frequency(c, 0.12) / 1e9).toFixed(2)} GHz`); // 2.5

// Period and angular frequency
function period(f) { return 1 / f; }
function angularFreq(f) { return 2 * Math.PI * f; }

// Sinusoidal wave: y(x, t) = A · sin(kx - ωt + φ)
function waveDisplacement(A, k, x, omega, t, phi = 0) {
  return A * Math.sin(k * x - omega * t + phi);
}

// String wave: A=2cm, k=2π/0.5 m, f=100 Hz
const k = 2 * Math.PI / 0.5;
const omega = 2 * Math.PI * 100;
console.log(`At x=0.1m, t=0.005s: y = ${waveDisplacement(0.02, k, 0.1, omega, 0.005).toFixed(4)} m`);

// Wave number from wavelength
function waveNumber(wavelength) { return 2 * Math.PI / wavelength; }

// Energy in mechanical wave (per unit length, on string of mass density μ)
function waveEnergyDensity(mu, omega, A) {
  return 0.5 * mu * omega * omega * A * A;
}

// String, μ = 0.01 kg/m, 100 Hz, A = 0.02 m
console.log(`Wave energy: ${waveEnergyDensity(0.01, omega, 0.02).toFixed(3)} J/m`);

Where wave properties matter

• Acoustics. Music tuning, speaker design, room acoustics, ultrasound imaging.
• Optics. Lens design, color, filters, imaging systems, lasers.
• Communications. Radio, TV, cell phones, fiber optics, satellite — all rely on wave physics.
• Medical imaging. Ultrasound, MRI, X-ray, CT, gamma cameras.
• Astronomy. Spectroscopy reveals frequencies → composition, temperature, motion of distant objects.
• Seismology. P, S, surface waves carry information about Earth's interior.
• Quantum mechanics. Matter waves, wave functions, fundamental physics descriptions.

Common mistakes

• Confusing wavelength with amplitude. Wavelength is the spatial period (distance between peaks). Amplitude is the height of peaks.
• Forgetting v = λf. Memorize this. Solve for any unknown given the other two.
• Confusing frequency with angular frequency. f in Hz, ω in rad/s. ω = 2πf. Use the right one in equations.
• Treating wave speed as fixed. v depends on the medium. Light in glass is slower than vacuum. Sound speed varies with T, density. Wave equation provides v from medium properties.
• Mixing transverse and longitudinal in formulas. Both have v=λf, but particle motion differs. Polarization only applies to transverse.
• Wrong scaling of energy. Mechanical wave energy ∝ A². Doubling amplitude → 4× energy, not 2×.